What's the difference between pressing S and Alt+S?
The key command S is used to scale whatever is selected in the viewport, regardless of mode. However using the command Alt+S to scale selection along the normals is only possible in Edit Mode.

When you scale the object, all the distance ratios between various vertices and all angles will remain exactly the same, just get larger or smaller:



However when using the Alt+S command, blender will try to shrink or fatten the mesh. This can lead to unusual results if done on certain meshes:



I have noticed that it works much better with more vertices. This really shows exactly what is happening. The inner vertices appear to have a greater weight, being moved a much greater distance. This leads to a convex shape after fattening, or a concave shape after shrinking.